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Chapter 2: Problem 57

Find equations of the tangent lines to the graph of \(f(x)=\frac{x+1}{x-1}\) that are parallel to the line \(2 y+x=6\). Then graph the function and the tangent lines.

Short Answer

Expert verified
The equations of the tangent lines to the graph of the function that are parallel to the line \(2y + x = 6\) are \(y = -0.5x -0.5\) and \(y = -0.5x + 3.5\).

Step by step solution

01

Find the derivative of the function

Firstly, you need to find the derivative of the function \( f(x) = \frac{x+1}{x-1} \). The derivative represents the slope of the tangent line at each point of the function. Use the quotient rule which states that \((d/dx)(u/v) = (vu'-uv')/v^2\), where \( u = x+1\) and \( v = x-1 \). The derivative \( f'(x) = \frac{(x-1)*1-(x+1)*1}{(x-1)^2} = \frac{-2}{(x-1)^2}\).
02

Determine the slope of the given line

The equation of the given line is in the slope-intercept form \(y=ax+b\), so we rewrite the equation to this form: \(2y+x=6\Rightarrow y = -0.5x + 3\). So the slope of this line \(a = -0.5\). Since parallel lines have equal slopes, the tangent line to the function must also have a slope of -0.5.
03

Find x-coordinates of the points of tangency

Next, find the x-coordinates of the points on the function's graph where the slope of the tangent line will be -0.5. Set the derivative equal to -0.5, and solve for x: \(-0.5 = \frac{-2}{(x-1)^2}\Rightarrow(x-1)^2 = 4 \Rightarrow x = 1 \pm 2\). Hence, there are two points: \(x = -1\) and \(x = 3\).
04

Find the equations of the tangent lines

The tangent line at any point \((x, f(x))\) on the graph of a function \(f\) can be found using the point-slope form of a line's equation: \(y - f(x) = f'(x)(x - x1)\), where \(x1\) is the x-coordinate of the point. Calculate function values at \(x = -1\) and \(x = 3\): \(f(-1) = \frac{-1+1}{-1-1} = 0\) and \(f(3) = \frac{3+1}{3-1} = 2\). Then, the equations of the tangent lines are (\(y - 0 = -0.5(x - (-1))\Rightarrow y = - 0.5x -0.5\) and \(y - 2 = -0.5(x - 3)\Rightarrow y = - 0.5x + 3.5\).
05

Graph the function and the tangent lines

Now, we will graph the function and the tangent lines using the obtained equations. This step will allow you to visualize the results and is typically done using graphing software or a calculator. The equation of the function is \(y = \frac{x+1}{x-1}\), and the equations of the tangent lines are \(y = -0.5x - 0.5\) and \(y = -0.5x + 3.5\).

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